Sheath Boundary Research Articles

Response to “Comment on ‘Dynamic behaviors of dust particles in the plasma–sheath boundary’ ” [Phys. Plasmas 9, 1057 (2002)

Abstract

Monte Carlo Simulations of the Influence of Ion-Neutral Collisions on the Ion Currents Collected by Electrostatic Probes

We have carried out Monte Carlo simulation of the motion of Ar + ions in the space charge sheath surrounding a cylindrical Langmuir probe. The ion currents to the probe have been calculated from these simulations and the percentages of ions crossing the sheath boundary that are collected by the probe have been determined. It has been shown that the collisions of ions with neutral helium gas atoms in the sheath increase the percentage of ions collected by the probe above that predicted by collisionless orbital motion limited current (OMLC) theory at lower helium pressure and decrease this percentage below the OMLC theory prediction at higher helium pressure. It has been shown also that the ion current almost does not depend on probe radius at higher helium pressures. The results of the simulations have been compared with recent Langmuir probe measurements made in flowing afterglow plasmas and with other probe theories.

Modeling of harmonic contributions to non-symmetrical RF plasmas

An analytical solution for the harmonic contributions to the dynamics of an asymmetrically driven capacitively coupled plasma is obtained under the assumptions of time-independent, collisionless ion motion, inertialess electrons and uniform current density. Modeling is performed considering the effect of an arbitrary Nth harmonic sinusoidal current. The overall effect is determined by summing up the effects due to all the current harmonics. The expressions for the instantaneous and time averaged electric field within the sheath, the non-linear oscillation motion of the electron sheath boundary, the effective sheath impedance, the bulk plasma impedance and finally the overall plasma impedance are obtained. Experimental results are shown to be in good qualitative agreement with the theoretical model.

Hybrid model for the calculation of ion distribution functions behind a direct current or radio frequency driven plasma boundary sheath

A hybrid fluid dynamic/kinetic model is presented which describes the sheath and the presheath regions of dc or rf driven low pressure gas discharges in a realistic and self-consistent way. The model assumes an infinite extended sheath parallel to the electrode, allowing a one-dimensional spatial description. It provides for the presence of multiple positive ion species and their collisional interactions with the neutral background, and takes into account the possibility of a nonharmonic modulation of the sheath potential and the application of an external dc bias; in this work, the model is applied to a two-species capacitively coupled argon and oxygen plasma. The input required by the model consists of the fluxes of the incoming ions, of the modulating current, and of the pressure, the composition, and the temperature of the background gas. On output, the model provides the values of the electric field and of the particle densities within the sheath and the presheath, the total voltage drop across the sheath, and also the energetically and angularly resolved distributions of the positive ions and the energetic neutrals which impinge the material substrate at the boundary. In general, the model is able to treat dc discharges as well as capacitively and/or inductively coupled rf discharges, it thus covers most of the plasmas used in very large scale integration microelectronics manufacturing and other surface modification techniques. Using the model, studies of the energy distributions of the incoming ions have been performed for a wide range of parameters, and the effects of varying process conditions have been investigated. At low and intermediate pressures (p<50 mTorr), the distribution functions of rf driven discharges exhibit a characteristic bimodal structure; this structure disappears with increasing pressure as ion-neutral collisions become significant. A comparison of calculated ion energy distributions with experimental measurements on capacitively coupled argon and oxygen discharges shows excellent quantitative agreement. In addition to the ion energy distribution, the angular distributions of the incident ions at various energies are also discussed as a function of the neutral gas pressure. It turns out that the details of the angular distribution not only depend on the field structure of the sheath itself but also on that of the presheath. The results of the presented model are therefore more reliable than those of previous models which restricted themselves to the sheath region. This high physical accuracy of the presented model, together with its flexibility and its high execution speed, allows its use as a tool for technology-oriented computer-aided design in the microelectronics industry.

Dynamic behaviors of dust particles in the plasma–sheath boundary

A variety of dynamic behaviors in dusty plasmas is expected under the experimental condition of weak friction with gas molecules. The device “KAGEROU” provides such an environment for dynamic collective phenomena. Self-excited dust oscillations in Coulomb crystals have been observed at low values of plasma density and gas pressure. An instability mechanism was identified to be delayed charging in an inhomogeneous equilibrium dust charge in the sheath. The theoretical growth rate was formulated in relation to the destabilization of a transverse dust lattice wave (T-DLW), which was found to be very sensitive to the presence of a small amount of hot electrons which produces a substantial positive equilibrium charge gradient ∇Qd-eq around the equilibrium position of dust particles in the plasma–sheath boundary. The first experimental observation of a correlated self-excited vertical oscillations in a one-dimensional dust chain indicates a destabilization of T-DLW. The experimental condition is very consistent with the parameter area which predicts numerically an instability of T-DLW.

Plasma etch/deposition modeling: A new dynamically coupled multiscale code and comparison with experiment

Next-generation plasma-process modeling tools can provide new insight into process dynamics by resolving the diverse length and time scales present in reactor systems. The length scales range from the size of the reactor (∼10 cm) to surface details (∼100 nm), and time scales from electron sheath-transit times (nanoseconds) to total process time (minutes). Other key features include dynamic coupling of the plasma and solid (particles and fields), and the ability to model realistic surface interactions (deposition, etch, sputter, polymerization, etc.). A computational tool has been developed which provides all of these features through the coupling of heterogeneous code modules [hybrid plasma, particle in cell (PIC) plasma and solid surface chemistry] and through time-sampling techniques. The hybrid code (particle ions, fluid electrons) provides the basis for modeling the large-scale plasma reactor using a finite-element mesh to represent complex reactor geometries. The PIC code is used in the dynamic sheath boundary region to account for electron movement. The solid surface chemistry code is specially developed to model complex interactions between surface mechanisms, such as the formation of polymer and its possible removal by high-energy particles. The solid surface module uses a finite element scheme with adaptive mesh refinement (on a many-cycle time scale) to follow the complex surface evolution. These code modules exchange information on a sub-rf-period timescale, allowing for direct solid/plasma interactions. The long process times (minutes) are simulated by result sampling and using the slow evolution of the plasma/solid system. The code also performs surface charge migration and local gas heating to more completely represent the physical processes occurring in a plasma processing operation. The modeling of plasma processing must also account for the multilayer films and many species of particles and material types. The incorporation of this information in the simulations has demonstrated mask erosion. The simulations performed using this code have also shown good correlation to experimental results for steady state etch rates, etch rates as a function of via size, sidewall polymer evolution, and the production of “sputter wind” and surface charging. In deposition mode, the simulations demonstrate the experimentally observed polymer and deposited film topology.

Influence of particulates generated in a CH4low pressure radiofrequency discharge on dc negative self bias voltage

The formation and the behavior of particulates have been studied in a pure methane rf plasma (13.56 MHz) for pressures in the range 20−150 Pa. The generated particulates are evidenced using laser light scattering. They are mainly observed at the sheath boundaries where they are suspended in well defined traps. Particulate appearance times at the upper sheath boundary have been determined using the 90° scattered light intensity at 514.5 nm. The appearance time of particulates is sensitive to the working pressure: the higher the pressure, the shorter the appearance time. In our experimental conditions, no particulates are observed in the cloud near the biased electrode for pressures lower than 30 Pa. The negatively charged particulates induce changes in the electrical characteristics of the discharge. The generation and the behavior of particulates are correlated with the time evolutions of the laser beam extinction and of the absolute value of the dc self bias voltage during an experiment.

Plasma Near a Divertor Plate: Effects of Surface Roughness and Particle Drifts

The surface of a divertor plate has usually a certain degree of ''roughness.'' Depending on the material and the exposure time, the size of surface features may range from submicrons to a fraction of a millimeter, covering a range of spatial scales from well below the electron gyroradius, {rho}{sub e} to significantly above the ion gyroradius, {rho}{sub i}. The plasma approaches the divertor plate along a magnetic field which forms a shallow angle, {alpha} << 1, with the plate surface. Under such circumstances, a significant ''shadowing'' effect takes place, with only a small part of the surface being accessible to the plasma particles. A methodology is presented that allows one to find the fraction ({var_epsilon}{sub e} and {var_epsilon}{sub i}) of the surface geometrically accessible for the electrons and the ions. At small {alpha}, {var_epsilon}{sub e,i} are typically small, meaning a strong local enhancement of heat and particle fluxes. In a broad range of parameters, {var_epsilon}{sub e}, is also much smaller than {var_epsilon}{sub i}. As the surface features are usually greater than the Debye radius, this leads to the formation of an ambipolar potential which causes reflection of part of the ions from the surface. The resulting albedo of the divertor plate for the plasma ions can be as high as 50%. Gradual diffusion of the plasma electrons into the zones reflecting the ions, reduces ambipolar fields and brings the ion albedo back to very small values. We present estimates of the time scales governing the neutralization process. This time scale shows itself up in the sheath boundary conditions for non-steady-state perturbations. The effect of surface roughness on secondary electron emission is discussed and it is shown that, depending on the surface structure, it may be smaller or greater than for a perfectly flat plate. We consider modifications of the sheath current-voltage characteristics by particle drifts. The presence of reflected ions reduces the ion diamagnetic current in the ion sub-sheath and significantly changes the ion response. This, in turn, affects sheath-controlled instabilities, which are sensitive to the tilt of the magnetic field.

Plasma Near a Divertor Plate: Effects of Surface Roughness and Particle Drifts

The surface of a divertor plate usually has a certain degree of “roughness”. Depending of the material and the exposure time, the size of surface features may range from submicrons to a fraction of a millimeter, covering a range of spatial scales from well below the electron gyroradius, ρi, to significantly above the ion gyroradius, ρi. The plasma approaches the divertor plate along a magnetic field which forms a shallow angle, α ≪ 1, with the plate surface. Under such circumstances, a significant “shadowing” effect takes place, with only a small part of the surface being accessible to the plasma particles. A methodology is presented that allows one to find the fraction (εe and εi of he surface geometrically accessible for the electrons and the ions. A small α, εe,i are typically small, meaning a strong local enhancemen of heat and particle fluxes. In a broad range of parameters, εe is also much smaller han εi. As the surface features are usually greater than the Debye radius, this leads to the formation of an ambipolar potential which causes reflection of part of the ions from the surface. The resulting albedo of the divertor plate for the plasma ions can be as high as 50%. Gradual diffusion of the plasma electrons into the zones reflecting the ions, reduces ambipolar fields and brings the ion albedo back to very small values. We present estimates of the time scales governing the neutralization process. This time scale shows itself up in the sheath boundary conditions for non-steady-state perturbations. The effect of surface roughness on secondary electron emission is discussed and it is shown that, depending on the surface structure, it may be smaller or greater than for a perfectly flat plate. We also briefly discuss modifications of the sheath boundary conditions by particle drifts.

Instability of a dusty plasma in the presence of a dc electric field and an equilibrium dust charge gradient

It is shown that a dusty plasma becomes unstable when a dc electric field and an equilibrium dust charge gradient are simultaneously present. Physically, the instability arises because the dc electric force as well as convection of the dust charge gradient produce dust charge fluctuations which cannot keep in phase with the potential of electrostatic fluctuations. This novel instability associated with the delayed dust charging can explain the salient features of that instability which is observed by Nonumura et al. [Phys. Rev. Lett. 83 (1999) 1970] near the sheath boundary in a dusty plasma.

Overexpression of Rice OSH Genes Induces Ectopic Shoots on Leaf Sheaths of Transgenic Rice Plants

Five rice homeobox (OSH) genes were overexpressed under the control of the cauliflower mosaic virus 35S promoter or the rice actin gene promoter in transgenic rice plants. Almost all of the transgenic plants showed abnormal phenotypes, which could be classified into three types according to their severity. Plants with the most severe phenotype formed only green organs, with many shoot apices on their adaxial sides. Plants with an intermediate phenotype formed bladeless leaves with normally developed leaf sheaths. Plants with a mild phenotype formed normal leaf sheaths and blades, but lacked ligules and showed diffusion of the blade–sheath boundary. The leaf structure of this phenotype was similar to that of dominant maize mutants, such as Kn1, Rs1, Lg3, and Lg4. Based on these phenotypes, we suggest that ectopic expression of the rice OSH genes interferes with the development of leaf blades and maintains leaves in less differentiated states. These results are discussed in relation to the leaf maturation schedule hypothesis (M. Freeling et al., 1992, BioEssays 14, 227–236).

Measurement of the interaction potential of microspheres in the sheath of a rf discharge

The interaction potential of two microspheres that are levitated in the sheath region of a radio frequency (rf) argon discharge is studied experimentally by analyzing their trajectories during head-on collisions. It is shown that the interaction parallel to the sheath boundary can be described by a screened Coulomb potential. Thus, values for an effective charge and a screening length can be obtained. The horizontal part of the interaction potential has been determined for several plasma conditions. There is no evidence for an attractive part in the potential within the accuracy of the present measurements and the given plasma conditions.

Simulation of plasma processes for microelectronic fabrication

Abstract

Non-linear sheaths with dust particles

Computer modeling of sheaths with dust particles, performed by means of the modified PIC method with self-consistent dust particle charge as well as electron and ion velocity distribution functions, show that dust particles can strongly influence the properties of these sheaths. This influence is caused by a selective collection and a scattering of electrons and ions by dust particles with self-consistent electric charge. Dust particles can strongly change an ion flux so that this flux is not constant in the sheath. That causes a change of both the well-known Bohms criterion on the sheath boundary and the other sheath parameter distributions.

A Conservative Finite Difference Method for the Numerical Solution of Plasma Fluid Equations

This paper describes a numerical method for the solution of a system of plasma fluid equations. The fluid model is similar to those employed in the simulation of high-density, low-pressure plasmas used in semiconductor processing. The governing equations consist of a drift-diffusion model of the electrons, together with an internal energy equation, coupled via Poisson's equation to a system of Euler equations for each ion species augmented with electrostatic force, collisional, and source/sink terms. The time integration of the full system is performed using an operator splitting that conserves space charge and avoids dielectric relaxation timestep restrictions. The integration of the individual ion species and electrons within the time-split advancement is achieved using a second-order Godunov discretization of the hyperbolic terms, modified to account for the significant role of the electric field in the propagation of acoustic waves, combined with a backward Euler discretization of the parabolic terms. Discrete boundary conditions are employed to accommodate the plasma sheath boundary layer on underresolved grids. The algorithm is described for the case of a single Cartesian grid as the first step toward an implementation on a locally refined grid hierarchy in which the method presented here may be applied on each refinement level.

Simple Hexagonal Coulomb Crystal near a Deformed Plasma Sheath Boundary in a Dusty Plasma

Simple hexagonal Coulomb crystals have been observed near a deformed plasma sheath boundary in a dusty plasma. It seems that particle rows, which are found in the direction perpendicular to an electrode in the Coulomb crystal, are formed due to not only Coulomb repulsive forces but also some attractive forces. In this study, experiments using the deformed sheath revealed that the rows were bent along the direction of electric fields and ion flows in a plasma. The result shows that sheath structures and shapes affect Coulomb crystal structures in dusty plasmas. In this paper, the formation of the particle rows are discussed, where balances of forces on particles in a dusty plasma are taken into account under the assumption that wake potentials related to ion flows exist in the sheath region.

Confinement and structure of electrostatically coupled dust clouds in a direct current plasma–sheath

Mechanisms for the confinement and the internal structure of an electrostatically coupled dust cloud formed in a dc glow discharge have been investigated from a comparative viewpoint between experimental observations and a simple model. Two kinds of dust clouds with different internal structures are clearly observed, depending on the dispersion of the size distribution of dust particles. The dust cloud can be trapped only in the plasma–sheath boundary area, corresponding to the potential minimum region determined by gravitational and electrostatic forces in the cathode sheath. No dust particles were found deep inside of the sheath, which is consistent with the analysis because the dust particles may be charged positively due to an extreme reduction of the electron density. The internal structure of the electrostatically coupled dust cloud was found to be arranged so that the total potential energy, including the repulsive Coulomb interaction among negative dust particles, may become minimal.

Arcjet Anode Plasma Measurements Using Electrostatic Probes

A 1-kW hydrazine arcjet thruster has been modie ed for internal probing of the anode sheath boundary layer with an array of 14 electrostatic microprobes e ush mounted into the anode body. Axial and azimuthal distributions of the plasma properties e oating potential, anode sheath potential, wall current density, electron number density, and electron temperature have been obtained for arc currents between 7.8 and 10.6 A and propellant e ow rates of 40 ‐60 mg/s. The specie c power ranged from 18.8 to 27.4 MJ/ kg. Azimuthal symmetry has been verie ed for all arcjet operating conditions. The electron temperature data show that the near-anode plasma is highly nonequilibrium. Most of the current density and anode heating is located within 2 ‐4 mm of the constrictor exit, with the location affected more by mass e ow rate than by arc current. The axial anode heating distribution is closely coupled to current density and accounts for ;18‐24% of the total input power. Reasonable agreement between a numerical model and experimental results is found for a constant value of the electron inelastic energy-loss factor.

Sheath criterion and boundary conditions for an electrostatic sheath

The effect of collisionality of ions on the formation of a positive sheath is investigated using a two-fluid model. It is the definition of the sheath edge that determines the sheath criterion. Bohm’s sheath criterion is based on the collisionless sheath and a quasineutral presheath. When collisions of ions in the sheath or finite space charge of presheath are taken into account, a sheath criterion can also be obtained based on the boundary conditions chosen at the sheath edge.

The maize gene liguleless2 encodes a basic leucine zipper protein involved in the establishment of the leaf blade-sheath boundary.

The blade and sheath of a maize leaf are separated by a linear epidermal fringe, the ligule, and two wedge-like structures, the auricles. In plants homozygous for the null mutation, liguleless2-reference (lg2-R), the ligule and auricles are often absent or positioned incorrectly and the blade-sheath boundary is diffuse. This phenotype is in contrast to that of liguleless1-reference (lgl-R) mutant plants, which have a more defined boundary even in the absence of the ligule and auricles. Additionally, mosaic analysis indicates the lg2-R phenotype is cell-nonautonomous and the lg1-R phenotype is cell-autonomous. Using scanning electron microscopy we show that lg2-R mutant plants are affected before the first visible sign of ligule and auricle formation. We have cloned the Lg2+ gene through a Mutator-8 transposon insertion allele, and verified it with five independently derived alleles. The comparison of genomic DNA and cDNA sequences reveals an open reading frame encoding a protein of 531 amino acids with partial homology to a subclass of plant basic leucine zipper (bZIP) transcription factors. Although a large body of molecular and biochemical characterization exists on this subclass of bZIP proteins, our work represents the first report of a mutant phenotype within this group. A specific reverse transcriptase (RT)-PCR assay shows LG2 mRNA expression in meristem/developing ligule regions. RT-PCR also shows that LG2 mRNA accumulation precedes that of LG1 mRNA. The mutant phenotype and expression analysis of lg2 suggest an early role in initiating an exact blade-sheath boundary within the young leaf primordia.